Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.


d. Compare the results from the critical value method, the P-value method, and the confidence interval method. Do they all lead to the same conclusion?

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that the proportion of zeros is equal to 0.1 (p = 0.1). The alternative hypothesis (H₁) states that the proportion of zeros is not equal to 0.1 (p ≠ 0.1).
2. Step 2: Calculate the sample proportion (p̂). The sample proportion is given by the formula p̂ = x / n, where x is the number of zeros (119) and n is the total number of digits (1000). Substitute the values into the formula to find p̂.
3. Step 3: Compute the standard error (SE) of the sample proportion. The formula for the standard error is SE = √((p * (1 - p)) / n), where p is the hypothesized proportion (0.1) and n is the sample size (1000). Substitute the values into the formula to calculate SE.
4. Step 4: Construct the confidence interval for the sample proportion. Use the formula p̂ ± Z * SE, where Z is the critical value corresponding to the desired confidence level (e.g., Z = 1.96 for a 95% confidence level). Substitute the values to determine the confidence interval.
5. Step 5: Compare the results from the critical value method, the P-value method, and the confidence interval method. For the critical value method, calculate the test statistic using the formula Z = (p̂ - p) / SE and compare it to the critical value. For the P-value method, find the P-value corresponding to the test statistic and compare it to the significance level (e.g., α = 0.05). For the confidence interval method, check if the hypothesized proportion (0.1) falls within the confidence interval. Determine if all methods lead to the same conclusion regarding rejecting or failing to reject the null hypothesis.

Key Concepts

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It provides an estimate of uncertainty around a sample statistic, such as a proportion. For example, if we calculate a 95% confidence interval for the proportion of zeros in the telephone numbers, it indicates that we can be 95% confident that the true proportion lies within this range.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (e.g., the proportion of zeros is 0.1) and an alternative hypothesis (e.g., the proportion of zeros is not 0.1). The results from tests, such as the critical value method and the P-value method, help determine whether to reject or fail to reject the null hypothesis based on the evidence provided by the sample.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, and if it is below a predetermined significance level (e.g., 0.05), we typically reject the null hypothesis.